Double factorial is a mathematical concept, using n! ! Express delivery. The double factorial of a positive integer represents all positive integer products that do not exceed this positive integer and have the same parity with it.

When n is a natural number, it means the product of all positive integers not exceeding n and having the same parity with n, such as:

Example:

3! ! = 1×3=3

5! ! = 1×3×5= 15

6! ! =2×4×6=48

8! ! =2× 4×6×8=384

Another 0! ! = 1! ! = 1

When n is a negative odd number, according to the recursive formula

, know n! ! The absolute value of is equal to the reciprocal of the absolute product of all negative odd numbers whose absolute values are less than their absolute values, and the positive and negative appear alternately. For example:

Example:

(-5)! ! = 1/(|- 1| × |-3|)= 1/3

(-7)! ! =- 1/(|- 1| × |-3| × |-5|)=- 1/ 15

(-9)! ! = 1/(|- 1| × |-3| × |-5| × |-7|)= 1/ 105

The other one (-1)! ! = 1

When n is a negative even number, we can know from the recursive formula that (-2)! ! =0! ! /0 is meaningless, so when n is a negative even number, n! ! Does not exist.

For positive integer n, there is (2n- 1)! ! (2n)！ ! =[ 1×3×…×(2n- 1)][2×4×…×(2n)]=(2n)！

For any integer n, there are